1 9 Graph and Solve Quadratic Inequalities FiveMinute

1. 9 Graph and Solve Quadratic Inequalities

Five-Minute Check CCSS Then/Now New Vocabulary Example 1: Graph a Quadratic Inequality Example 2: Solve ax 2 + bx + c < 0 by Graphing Example 3: Solve ax 2 + bx + c ≥ 0 by Graphing Example 4: Real-World Example: Solve a Quadratic Inequality Example 5: Solve a Quadratic Inequality Algebraically

Write y = 5 x 2 + 30 x + 44 in vertex form. A. y = x 2 + 6 x + 11 B. y = 5(x + 3)2 – 1 C. y = 5(x – 3)2 D. y = (x + 3)2 + 1

Identify the vertex of y = 5 x 2 + 30 x + 44. A. (3, 1) B. (– 1, – 3) C. (– 2, – 1) D. (– 3, – 1)

Find the axis of symmetry of y = 5 x 2 + 30 x + 44. A. x = 3 B. x = 0 C. x = – 2 D. x = – 3

What is the direction of the opening of the graph of y = 5 x 2 + 30 x + 44? A. upward B. downward C. right D. left

Graph the quadratic function y = 5 x 2 + 30 x + 44. A. B. C. D.

The graph of y = x 2 is reflected in the x-axis and shifted left three units. Which of the following equations represents the resulting parabola? A. y = (x – 3)2 B. y = –(x – 3)2 C. y = –(x + 3)2 D. y = –x 2 + 3

Content Standards A. CED. 1 Create equations and inequalities in one variable and use them to solve problems. A. CED. 3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Mathematical Practices 1 Make sense of problems and persevere in solving them.

You solved linear inequalities. • Graph quadratic inequalities in two variables. • Solve quadratic inequalities in one variable.

• quadratic inequality

Graph a Quadratic Inequality Graph y > x 2 – 3 x + 2. Step 1 Graph the related quadratic equation, y = x 2 – 3 x + 2. Since the inequality symbol is >, the parabola should be dashed. >

Graph a Quadratic Inequality Step 2 Test a point inside the parabola, such as (1, 2). y > x 2 – 3 x + 2 ? 2 > 12 – 3(1) + 2 ? 2>1– 3+2 2>0 So, (1, 2) is a solution of the inequality. >

Graph a Quadratic Inequality Step 3 Shade the region inside the parabola that contains the point (1, 2). Answer: >

Which is the graph of y < –x 2 + 4 x + 2? A. B. C. D.

Solve ax 2 + bx + c < 0 by Graphing Solve x 2 – 4 x + 3 < 0 by graphing. The solution consists of the x-values for which the graph of the related quadratic function lies below the x-axis. Begin by finding the roots of the related equation. x 2 – 4 x + 3 = 0 Related equation (x – 3)(x – 1) = 0 Factor. x – 3 = 0 or x – 1 = 0 x =3 x =1 Zero Product Property Solve each equation.

Solve ax 2 + bx + c < 0 by Graphing Sketch the graph of the parabola that has x-intercepts at 3 and 1. The graph should open up since a > 0. The graph lies below the x-axis to the right of x = 1 and to the left of x = 3. Answer: The solution set is {x | 1 < x < 3}.

What is the solution to the inequality x 2 + 5 x + 6 < 0? A. {x | – 3 < x < – 2} B. {x | x < – 3 or x > – 2} C. {x | 2 < x < 3} D. {x | x < 2 or x > 3}

Solve ax 2 + bx + c ≥ 0 by Graphing Solve 0 ≤ – 2 x 2 – 6 x + 1 by graphing. This inequality can be rewritten as – 2 x 2 – 6 x + 1 ≥ 0. The solution consists of the x-values for which the graph of the related quadratic equation lies on and above the x-axis. Begin by finding roots of the related equation. – 2 x 2 – 6 x + 1 = 0 Related equation Use the Quadratic Formula. Replace a with – 2, b with – 6, and c with 1.

Solve ax 2 + bx + c ≥ 0 by Graphing or Simplify and write as two equations. Simplify. Sketch the graph of the parabola that has x-intercepts of – 3. 16 and 0. 16. The graph should open down since a < 0. Answer: The graph lies on and above the x-axis at x = 0. 16 and x = – 3. 16 and between these two values. The solution set of the inequality is approximately {x | – 3. 16 ≤ x ≤ 0. 16}.

Solve ax 2 + bx + c ≥ 0 by Graphing Check Test one value of x less than – 3. 16, one between – 3. 16 and 0. 16, and one greater than 0. 16 in the original inequality. Test x = – 4. 0 ≤ – 2 x 2 – 6 x + 1 ? Test x = 0. 0 ≤ – 2 x 2 – 6 x + 1 ? 2 0 ≤ – 2(– 4) – 6(– 4) + 1 0 ≤ – 2(0)2 – 6(0) + 1 0 ≤ – 7 0≤ 1 Test x = 1. 0 ≤ – 2 x 2 – 6 x + 1 ? 0 ≤ – 2(1)2 – 6(1) + 1 0 ≤ – 7

Solve 2 x 2 + 3 x – 7 ≥ 0 by graphing. A. {x | – 2. 77 ≤ x ≤ 1. 27} B. {x | – 1. 27 ≤ x ≤ 2. 77} C. {x | x ≤ – 2. 77 or x ≥ 1. 27} D. {x | x ≤ – 1. 27 or x ≥ 2. 77}

Solve a Quadratic Inequality SPORTS The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by the function h(x) = 40 x – 16 x 2, where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 15 feet of the ground? The function h(x) describes the height of the ball. Therefore, you want to find values of x for which h(x) ≤ 15 Original inequality 40 x – 16 x 2 ≤ 15 – 16 x 2 + 40 x – 15 ≤ 0 h(x) = 40 x – 16 x 2 Subtract 15 from each side.

Solve a Quadratic Inequality Graph the related function – 16 x 2 + 40 x – 15 = 0 using a graphing calculator. The zeros are about 0. 46 and 2. 04. The graph lies below the x-axis when x < 0. 46 or x > 2. 04. Answer: Thus, the ball is within 15 feet of the ground for the first 0. 46 second of its flight, from 0 to 0. 46 second, and again after 2. 04 seconds until the ball hits the ground at 2. 5 seconds.

SPORTS The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function h(x) = 28 x – 16 x 2, where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 10 feet of the ground? A. for the first 0. 5 second again after 1. 25 seconds B. for the first 0. 5 second only C. between 0. 5 second and 1. 25 seconds D. It is never within 10 feet of the ground.

Solve a Quadratic Inequality Algebraically Solve x 2 + x ≤ 2 algebraically. First, solve the related quadratic equation x 2 + x = 2 Related quadratic equation x 2 + x – 2 = 0 Subtract 2 from each side. (x + 2)(x – 1) = 0 Factor. x + 2 = 0 or x – 1 = 0 x = – 2 x=1 Zero Product Property Solve each equation.

Solve a Quadratic Inequality Algebraically Plot – 2 and 1 on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 intervals. Test a value in each interval to see if it satisfies the original inequality.

Solve a Quadratic Inequality Algebraically Answer: The solution set is {x | – 2 ≤ x ≤ 1}. This is shown on the number line below.

Solve x 2 + 5 x ≤ – 6 algebraically. What is the solution? A. {x | – 3 ≤ x ≤ – 2} B. {x | x ≥ – 2 or x ≤ – 3} C. {x | 1 ≤ x ≤ 6} D. {x | – 6 ≤ x ≤ – 1}